A) \[\sqrt{52}\]
B) \[\sqrt{5}\]
C) \[{{G}^{1/2}}{{h}^{1/2}}{{c}^{-5/2}}\]
D) \[{{G}^{-1/2}}{{h}^{1/2}}{{c}^{1/2}}\]
Correct Answer: A
Solution :
Key Idea According to the method of dimensional analysis the dimension of each term on both sides of an equation must be same. \[dB=10{{\log }_{10}}\left[ \frac{I}{{{I}_{0}}} \right],\] \[{{I}_{0}}={{10}^{-12}}w{{m}^{-2}}\] \[4-=10{{\log }_{10}}\left[ \frac{{{I}_{1}}}{{{I}_{0}}} \right]\] Putting the dimensions in above relation \[\Rightarrow \]\[\frac{{{I}_{1}}}{{{I}_{0}}}={{10}^{4}}\] \[20=10\log {{ & }_{10}}\left[ \frac{{{I}_{2}}}{{{I}_{0}}} \right]\]\[\Rightarrow \] Comparing the powers of M, L and T \[\frac{{{I}_{2}}}{{{I}_{0}}}={{10}^{2}}\] ...(i) \[\Rightarrow \] ...(ii) \[\frac{{{I}_{2}}}{{{I}_{1}}}={{10}^{-2}}=\frac{r_{1}^{2}}{r_{2}^{2}}\] ...(iii) On solving Eqs. (i), (ii) and (iii) \[\Rightarrow \] Hence, dimensions of time are \[r_{2}^{2}=100_{1}^{2}\]
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